Spatiotemporal Analysis of Tuberculosis using Hausdorff–Gaussian Processes
Available at lcgodoy.me/slides/2024-cobal/
2024-12-05
TB in Context: The United Nations’ Millennium Declaration in 2000 outlined eight development goals to be achieved by 2015, including combating Tuberculosis (TB).
Mortality: TB remains a major global health threat, second in infectious disease mortality only to COVID-19.
Brazilian TB Trends: Brazil’s national TB incidence declined only slightly from 2006–2015.
Rio Grande do Sul (RS) reported significantly higher incidence than the national average in 2021, with the eastern region even more affected.
Spatial Dependence: Studies demonstrate strong spatial dependence of TB infections in Brazil, both nationwide and in specific cities.
Temporal dependence has been largely overlooked in the literature.
Risk Factors: TB risk factors include densely populated areas, poverty, substance abuse, and incarceration (Cortez et al. 2021).
Spatial models: CAR (Besag 1974), ICAR, BYM (Besag et al. 1991), DAGAR (Datta et al. 2019), RENeGe (Cruz-Reyes et al. 2023).
Nonseparable SPT models are more complex as they consider that the spatial and temporal correlations might be intertwined (Cressie and Wikle 2015, pg. 309–321).
Separable models one way to look at these models is as multivariate spatial processes (MacNab 2022).
Advantages of separable models: Computational efficiency & positive-definiteness of the covariance function.
Hausdorff–Gaussian Process (HGP): we propose using the newly developed HGP for the spatial portion of the model (Godoy et al. 2024).
Reliable incidence estimates:
Forecasting: Predicted TB incidence rates one year ahead offer crucial insights for proactive public health planning.
Areal spatial units are (closed and bounded) sets.
We need to generalize distance between points to distance between sets.
Ideally, this distance should:
Distance between a point and a set: \(d(x, A) = \inf_{a \in A} d(x, a)\), where \(d(x, y)\) is the distance between any two elements \(x, y \in D\)
Directed Hausdorff & Hausdorff distance: \[{\vec h}(A, B) = \sup_{a \in A} d(a, B) \quad \text{and} \quad h(A, B) = \max \left \{ \vec{h}(A, B), \vec{h}(B, A) \right \}\]
General spatial model: \(\{ Z(\mathbf{s}) \; : \; \mathbf{s} \in \mathcal{B}(D) \}\).
Index set: \(\mathcal{B}(D)\) represents the closed and bounded subsets of \(D \subset \mathbb{R}^2\).
Assumption: The HGP assumes \(Z(\mathbf{s})\) to be an isotropic Gaussian Process such that its spatial correlation function depends on the Hausdorff distance.
Powered Exponential Correlation (PEC) function: \(r(h) = \exp\left \{ - \frac{h^{\nu}}{\phi^{\nu}}\right \},\) where \(h\) denotes the Hausdorff distance between \(\mathbf{s}_1, \mathbf{s}_2 \in \mathcal{B}(D)\).
A generalized linear mixed effects model (GLMM) can be written as \[\begin{aligned} & Y_t(\mathbf{s}_i) \mid \mathbf{x}_{it}, Z(\mathbf{s}_i, t) \overset{{\rm ind}}{\sim} f(\cdot \mid \mu_{it}, \boldsymbol{\gamma}) \\ & g(\mu_{it}) = \alpha + \mathbf{x}_{it} \boldsymbol{\beta} + Z(\mathbf{s}_i, t). \end{aligned}\]
Probability distribution: \(f(\cdot)\)
Conditional mean: \(\mu_{it} = \mathbb{E}[Y_t(\mathbf{s}_i) \mid \mathbf{x}_i, Z(\mathbf{s}_i, t)]\)
Link function: \(g(\cdot)\)
Model parameters: \(\boldsymbol{\theta} = {\{\boldsymbol{\beta}^\top, \boldsymbol{\sigma}^\top, \boldsymbol{\delta}^\top, \boldsymbol{\gamma}^\top \}}^\top\)
Joint density: \(p(\mathbf{y} \mid \mathbf{z}, \boldsymbol{\theta}) = \prod_{i = 1}^n f(y_t(\mathbf{s}_i) \mid \mu_{it}, \boldsymbol{\gamma})\)
We assume \(Z(\mathbf{s}, t)\) is a separable zero-mean GP with a covariance function that is the product between the HGP covariance function and the \(\mathrm{AR}(1)\) covariance function.
Spatial dependence: \(\rho \sim \mathrm{Exp}(a_\rho)\), where \(a_{\rho} = - \log(p_{\rho}) / \rho_0\). \(a_\rho\) is chosen such that \(\mathbb{P}(\rho > \rho_0) = p_\rho\).
Smoothness & marginal SD: \(\nu \sim \mathrm{Beta}(2.5, 1.5)\) (mode at \(0.75\)) & \(\sigma \sim t_{+}(3)\).
Temporal dependence: PC prior (Sørbye and Rue 2017) where \(\mathbb{P}(\lvert \psi \rvert > 0.8) = 0.1\).
Intercept & regression coefficients: \(\alpha\) (i.e., \(\pi(\alpha) \propto 1\)) & \(\boldsymbol{\beta} \sim \mathcal{N}(\mathbf{0}, 10 \mathbf{I})\)
Super naive: \(vec(\mathbf{Z}) \sim \mathcal{N}(\mathbf{0}, \sigma^2 \mathrm{R}_s \otimes \mathrm{R}_t)\) requires \(\mathcal{O}(N^3 T^3)\) flops (and storage).
Naive: With linear algebra, we can reduce the computational complexity (and storage) to \(\approx \mathcal{O}(N^3 + T^3)\)
Astute: More linear algebra can be used to evaluate a quadratic form with less operations.
Clever: The Cholesky decomposition of \(R^{-1}_t\) is tridiagonal.
Super clever: The complexity to obtain \(chol(R^{-1}_s)\) is dramatically decreased using nearest-neighbor approximations (Finley et al. 2019).
Posterior: \(\pi(\boldsymbol{\theta} \mid \mathbf{y}, \mathbf{z}) \propto p(\mathbf{y} \mid \mathbf{z}, \boldsymbol{\theta}) p(\mathbf{z} \mid \boldsymbol{\theta}) \pi(\boldsymbol{\theta})\)
MCMC sampler: No-U-Turn (Homan and Gelman 2014).
Convergence assessment: traceplots and split-\({\hat{R}}\) (Vehtari et al. 2021).
Goodness-of-fit criteria: LOOIC (lower values indicate better fit)
Posterior predictive distributions: \(p(\mathbf{y}^{\ast} \mid \mathbf{y})\)
Predictions assessment: Interval Score (IS) and RMSP (lower values indicate better fit)
Sample units: 54 municipalities, across 11 years (2011 to 2021). We use 2022 to assess the quality of predictions.
Number of TB cases: \(Y_t(\mathbf{s}_i)\) at location \(\mathbf{s}_i\) and time \(t\).
Population: \(P_t(\mathbf{s}_i)\).
Five covariates and two way interactions with presence of prison (except IDESE).
\[\begin{aligned} & (Y_t(\mathbf{s}_i) \mid \mathbf{X}_{t}(\mathbf{s}_i), Z(\mathbf{s}_i, t)) \overset{{\rm ind}}{\sim} \text{Poisson}(P_t(\mathbf{s}_i) \mu_{it}) \\ & \log(\mu_{it}) = \alpha + \mathbf{X}^\top_{t}(\mathbf{s}_i) \beta + Z(\mathbf{s}_i, t) \end{aligned}\]
| LOOIC | RMSP | IS | |
|---|---|---|---|
| HGP | 3516.1 | 21.1 | 87.8 |
| BYM | 3606.1 | 123.3 | 176.6 |
| DAGAR | 3520.9 | 22.4 | 88.8 |
| Parameter | Description | Estimate |
|---|---|---|
| \(\exp(\beta_1)\) | Prison | 2.34 (1.70, 3.19) |
| \(\exp(\beta_2)\) | Pop / km2 | 1.33 (1.15, 1.56) |
| \(\exp(\beta_2 + \beta_{21})\) | 1.75 (1.18, 2.52) | |
| \(\exp(\beta_3)\) | HS dropout % | 1.03 (0.99, 1.07) |
| \(\exp(\beta_3 + \beta_{31})\) | 2.25 (1.63, 3.09) | |
| \(\exp(\beta_4)\) | Homicide rate | 0.97 (0.93, 1.00) |
| \(\exp(\beta_4 + \beta_{41})\) | 2.51 (1.83, 3.46) | |
| \(\exp(\beta_5)\) | IDESE | 0.99 (0.92, 1.07) |
Tailored an HGP extension for spatiotemporal disease mapping.
Competitive with specialized models
It helps to gain insights into spatiotemporal disease mapping through spatiotemporal correlation functions.
More reliable estimates of risk factors
Out-of-sample predictions to inform public policies